How to keep shape of geometry when adjusting mesh?
Whenever you have an object you want to smooth by adding geometry you should always try to work solely with quads (which you are doing right). However, vertexes which have more than 4 edges connected will result in an uneven smooth.

Topology like the image below is better but still not perfect. It really depends on what your end use is.
You, can add more or less sharpness by inserting 'edge loops' closer or further from the borders (which only works with good topology).


A better way going about modeling this would be something like the following image: (not perfect but should help you get on the right path)